


Overview
General Formal Ontology (GFO)

Subsections
2.4 Translations and Ontological Mappings
The comparison of ontologies assumes a notion of semantic
transformation and ontological
mapping.
Let
be
an ontology and
; we say that a sentence from
is ontologically compatible with if is
consistent with . A semantic mapping (or semantic transformation) of an ontology
into the ontology
is a computable function
such that
. The most important semantic mappings are
interpretations in the sense of logic and model theory (59).
We sketch the main ideas of the method of interpretability in the
framework of theories in firstorder logic (cf. (58)). A theory is said to be
interpretable in the theory if it is obtainable from by means of definitions. The question is
which schemas of definitions are admitted, and what  in general  a
definition is. The simplest case of definitions are explicit
definitions which are assumed in the sequel. Let us assume that and
are theories in the (firstorder) languages , and ,
respectively. Translations from into are defined by means of
codes. A code in the sense of (58)  in the simplest case
 has the form
, where ,
, ..., are formulas of the language specified
in the vocabulary ; here, a formula is associated to every
relation symbol , such that the arity of
equals the number of free variables of . The formulas
serve as explicit definitions of the relational symbols . A
translation from into associates to every formula of
a formula of . Translations based on a code are recursively
defined (for details, see (58)).
A theory is said to be (syntactically) interpretable in if 
which is based on the code  satisfies the following condition:
(C) For every sentence
holds:
if and
only if
.
Generally, a theory is interpretable in if a code
exists such
that the translation which is based on satisfies condition
(C). Note that codes can be much more complicated than the simple
version mentioned above.
In modelling a concrete domain we start with a body of
source information about , denoted by , which is usually
presented in different languages (including natural language), often
in a nonstructured form. From a specification (which
takes the form of a set of expressions) is constructed with the aim to
capture the knowledgecontent of . Usually, is expressed
in a (formal) modeling/representation language, but also in natural or
semiformal languages, here denoted by ML (modeling language). Examples of
such languages are: KIF (22), Description Logics
(4),
Conceptual Graphs (53), and Semantic Networks, but also
modeling languages like UML (Unified Modeling Language)
(48)
or OPM (Object
Process Methodology) (20). In general, the
system is not sufficiently ontologically founded, and it
remains the task to translate it into an ontologically founded and
formal knowledge base which is formulated in some target
language (Ontology Language). An ontological
mapping
translates the expressions of into the language resulting
in the knowledge base
, which captures formally the
ontological content of . We say, in this case, that
is an ontological foundation of .
We explain the notion of ontological mapping for
terminology systems. In general, a terminology
system
consists of a language ,
a set of concepts, a set of relations between these concepts,
and a
function which associates to every concept or relation
a definition
which is an expression of the language .
Let
be a terminology
system and
an (formalized) ontology called a
reference
ontology for . An ontological
mapping from
into is a (partial) function from
into such that for
every concept in the expressions and are
semantically equivalent with respect to . In this case we may define
a formal knowledge base
which explicitly extracts the content in and
provides inference mechanisms. Note, that is in general not a
foundational ontology, but we assume that is constructed from
a foundational ontology, say GFO, by a number of welldefined steps. A detailed discussion of this
method is presented in (31).
Robert Hoehndorf
20061018

